byAngel Johnson

# Rationale

"The true virtue of mathematics (and not many know this) is that it saves efforts". This sadly underappreciated truth, stated so eloquently by Ron Aharoni, is the main premise of this unit. According to Aharoni, mathematics has three ways to economize: order, generalization, and concise representation. He refers to these as mathematical economy. Order is discerning patterns to make orientation easier; generalization is the idea that an idea discovered in one area can be applied to another area; and concise representation refers to mathematical formulas that represent propositions in a brief and clear manner. In this unit, all three of the economies will be addressed; however, concise representation will be the primary focus. The basis of concise representation allows us to translate our verbal and written words into a mathematical representation. As many teachers and students are very aware - "mathematics is a language of itself" that uses numbers as well as words. Aharoni defined this as a "mathematical proposition" - the equivalent of a sentence in spoken language. We are very familiar with formulas which are precise mathematical propositions that were condensed. By understanding and perfecting the art of translating English sentences to algebraic expressions, and vice versa, students will be better equipped to deal with the "infamous" word problems.

As discussed among my colleagues within my school as well as in the seminar, many students enjoy working with numeric mathematics problems as opposed to word problems for a number of reasons. Early in their education students become familiar with numerical expressions. Often those expressions are limited to addition and subtraction. Generally, they are very concrete. By the middle grades, students begin to study multiplication and division expressions with limited abstraction attached. Unfortunately, the tendency is to study computation only, to the neglect of applications through word problems. Therefore, by the time students arrive in high school, specifically Algebra, they have little experience in interpreting expressions or applying them in word problems. Needless to say, students prefer to avoid word problems and are uncertain of what is being asked and how to approach the word problem. To help students to reach a level of understanding of solving word problems, students must be introduced to word problem by a systematic approach. This unit will explore approaches to, and ultimately the solution of word problems through the use of numerical relationships and variables. Students will investigate, describe, and generalize a variety of patterns and represent situations using variables. They will also simplify algebraic expressions involving like terms and when given values are provided. Students will also review properties of numbers and operations using variables.

To reduce frustration and allow for "success", many teachers tend to stay away from word problems; however, word problems must not be ignored. Word problems must be an integral part of the curriculum. I generally try to use curricula and lessons that are relevant as well as familiar, as I will do in this unit. This will increase students' interest and achievement. In fact, Aharoni states, "one of the fundamental principles of teaching is to begin with the familiar." For example, students are familiar with reading; they may not be proficient with it, but the structure already exists in their mind. Therefore, adding word problems is an extension of reading curricula. Additionally, one principle for consistent success in solving word problems is that it is absolutely necessary to read the problem carefully, if necessary, read it again. A third reading may also be needed. Additionally, embedded into reading is the use of common language. Similarly, in math, along with all other subjects, it is imperative that there is a common language. However, the challenge is that there are many words that are used interchangeably. If their equivalence is not recognized, this will cause confusion. For the purpose of this unit, some of the words that present this challenge and may need clarification are listed in appendix 1.

Although, many teachers feel that teaching word problems is a "separate" unit, the integration of word problems with numerical problems will deepen student understanding as well as problem solving skills. The problem that I have with this integration is that many of my ninth grade Algebra students enter my class unequipped with the essential mathematic skills necessary to begin the algebra curriculum, or better yet, to be able to tackle word problems. In past years, students that were below the 48th percentile of the State's test were automatically enrolled in pre-algebra to develop weak skills. Today, the district now requires that all freshman students take Algebra as their first high school mathematics course - regardless of developmental level. This poses a tremendous challenge to all algebra teachers in my district as well as the students. In order to lessen the anxiety and frustration of my students, I constantly look for new ways to help my students achieve by staying abreast with the best instructional strategies and incorporating them whenever I feel they will help my students.

In addition to the above, my school is the second lowest high school in the city in regard to the Prairie State Achievement Examination (PSAE) scores. This summer our reform-minded principal who was hand-picked by school officials was transferred out the building. Prior to his principalship, the school was in physically bad shape; a third of the computers worked, showers were inoperable, the band had no uniforms, and more than a third of the students received special education services. Under the new principal's guidance, many strategies were implemented and renovation was made: the schools received new computers and books as well as a renovated swimming pool. Unfortunately, the improvements did not result in significant student academic gains. In fact, performance and attendance dropped in the last few years. In 2003, 5.9 percent students were passing the PSAE tests. In 2005, the number had dropped to 3.8 percent. Since 2003, the schools' four-year graduation rate has fallen almost 10 percentage points to 48.7. The attendance rate has also dropped. One area of slight improvement occurred with the ACT. The ACT scores have improved, rising from 13.9 in 2003 to 14.4 in 2006 on a 36 point-scale.

The Algebra curriculum involves an approach that encourages students to work in groups. Most of the activities are discovery-based cooperative learning. The idea is that students will make connections and take ownership of the lessons. The problem is that most of my students are reluctant to work in groups and prefer that the "rules" or shortcuts be given to them. A second problem is that the curriculum does not provide opportunity for the students to master the skills before beginning the next unit. In addition, the curriculum provides only limited opportunity to work with "traditional" word problems. Although the curriculum supports my belief that students must make connections as well as discoveries, I also strongly believe they need more exposure to reading, analyzing, and solving word problems.

Algebra I is the study of linear equations and functions. Throughout this course, students will examine linear equations, graphs, and tables. They will apply the skills that were developed to solve real life situations that usually come in word problems. It will be imperative that students know how to translate verbal expressions to algebraic expressions - the problem territory - and ultimately solve word problems. There are three dimensions of the problem territory: understand the relationship of the variable and the operation symbols, represent the variables and operations correctly, and evaluate expressions given a value of the variable. Despite the many challenges my students and I face, my goal for the completion of this unit is to have student master the state's standard of formulating and evaluating linear expressions algebraically. This goal is important because these are the fundamental skills for future progress.

For most students, math means that they will have to work with numbers, variables, operations and expressions. The general principle of the unit problem sets will enable the students to solve set of word problems that require writing expressions. However, the problems are carefully graduated in their complexity as well as the complexity of the required operation(s). Initially, as students explore the problem territory, they will use arithmetic that will extend to algebraic representation through the concrete, pictorial, and abstraction methods. They will begin to develop a deeper understanding in writing expressions and evaluating word problems. Students will begin to recognize that there are many approaches to solving mathematics problems. The opportunity for reinforcement and enhancement will be provided. By using problem solving strategies, students will organize information and solve novel as well as unique problems.

# Strategies

Overall, my students have very difficult time processing and retaining information. Therefore, the unit will incorporate scaffolding the concepts. This will allow students to review prior knowledge and build on it. The unit will start with students describing and modeling linear expressions and then move to students evaluating word problems. Students will be engaged using individual, small collaborative groups, and whole class discussions as well as cooperative learning. Through the interactions, we will discuss the various strategies for writing and interpreting expressions, and translating word problems into symbolic language. Then students will complete a K-W-L assessment chart. K-W-L is an assessment strategy used to adjust instruction to individual needs. It is a visual reading strategy that has been incorporated in other subjects. Teachers activate students' prior knowledge by asking them what they already Know; then students (collaborating as a classroom unit or within small groups) set goals specifying what they Want to learn; and after (reading/ discussing) students discuss what they have Learned (KWL website). Students will apply higher-order thinking strategies which help them construct meaning from what they have read/discussed. This will help them monitor their progress toward their goals. A KWL worksheet that includes columns for each of these activities will be provided to every student.

Additionally, I will use the Singaporemath Curriculum to develop and deepen student understanding by using many of its visual/graphic word problems. The Singapore curricula have been very successful in developing student understanding of word problems and ensuring mastery of important skills. It incorporates the principle of moving from concrete, to pictorial, and then to abstract representations of ideas. It does not focus on a large range of topics instead limits the topics that are taught within each grade. The focus is on mastery and deepening student understanding. It also helps students make connections between pictures, words, and numbers. Scaffolding and mental math strategies are embedded in the curriculum. Essentially, it is a visually-based math program that benefits diverse learners.

I will also incorporate whole group discussions. The whole group format creates opportunities to clarify common misunderstandings and personifies a culture of learning which incorporates every student's involvement in a collective effort of understanding. This format will provide opportunities to generate productive mathematical dialogues among students with different understanding levels, and also opportunities for applying the concepts. Students will be able to use previous knowledge as well as newly acquired knowledge of translating verbal expressions. They will also make connections and add personal experiences to more complex word problems. It also will allow students to share their ideas in order to raise their curiosity and participation in learning. As students describe their interpretations of the word problems to the class, they can clarify their thinking as well as their classmates'. Another student's approach can supply a new perspective, which tends to produce flexible thinking. All students will have the opportunities to add to the discussions. It also will encourage students to develop their language skills, both in math and in everyday English. It will enable students to become active listeners and questioners, creating multidirectional student-to-student talk that stimulates engagement and community.

Throughout this unit, students will be encouraged to use the TI-83 graphing calculator for support and reinforcement. The TI-83 is a very integral tool in evaluating expressions. It serves as a visual and "hands-on" manipulative that helps students reinforce their skills; and it saves time. Students will participate in a calculator workshop learning the features of the "equation solver" as an extension activity.

In addition, the nine properties of real numbers (rule of arithmetic) is used as well as reviewed. A list of the nine rules is outlined in the appendix. This unit is designed to use as an extension lesson. It is expected that students are able to use mental math effectively and have some knowledge of solving equations. The goal is to take that knowledge and apply it to solving word problems.

# Ice -Breaker - I

Students will work in small collaborative groups and create a list of "common language words" for addition, subtraction, multiplication, and division. The whole class will reconvene and through discussions each group will add any words that are omitted from their list. Some common language words have different meanings depending on the content of the questions. To ensure that students understand this, the following exercise will be presented. The emphasis will not be on solving the problems but the operations and wording.

Problem 1:

• a) Andy has 38 videotapes. His friend Roger has 27 videotapes. How many more video tapes does Andy have than Roger? (11 videotapes)
• b) Toni has 21 beads. Her friend Zora has 6 beads. How many more beads does Toni have than Zora? (15 beads)

Generally, the word "more" suggests addition. However, in this problem set the operation is subtraction.

Problem 2:

• a) A watch costs \$167. A camera costs \$48 more than the watch. What is the cost of the camera? (\$215)
• b) My old pogo-stick record was 349 jumps. Today I made 532 more jumps than that. What's my new Pogo-stick record? (881 jumps)

In these problems, the word "more" implies addition. Appendix II has more problems that address this issue.

# Lesson One - Notation Development (whole class discussion)

In order for students to be successful in solving word problems, they must become comfortable in working with numerical expressions as well as algebraic expressions. They should evaluate complex numerical expressions, especially those that will be similar in word problems. We want them to be able to write and verbalize expressions. Numerical and algebraic expressions are used throughout algebra. Numerical expressions contain all numbers whereas algebraic expressions contain numbers and variables.

Reading and translating the meaning of algebraic notations are therefore essential skills. Students should be able to convert verbal descriptions given in word problems into numerical expressions, and be able to read and evaluate numerical expressions including parentheses and nested parentheses. In this unit, we only intend to develop linear equations, so issues of exponents will not arise.

Numerical Expressions

Basic:

4+ 4

2*3

12-9

-11-4

24/6

Complex:

(2+4)5 - 7(6-1)

18 (2 + (12 /3))

(24 /3-2) + 4(12x - 3)

4(2x (7-3) + 5x) - 5 + 3/4

Ask the students to provide more examples.

Algebraic Expressions

As stated earlier, concise representation is one of the ways mathematics saves work. It uses letters and numbers instead of words. An algebraic expression is a mathematical phrase that consists of one or more numbers and variables along with one or more arithmetic operations. In the algebraic expression 6x, the letter x is called a variable and 6 is the constant. It is also called the coefficient of x. Variables are symbols used to represent unspecified numbers or values. Any letter may be used as a variable. (This example is to show students how algebraic expressions look. At this point, students may not know how to convert verbal descriptions to expressions and know how to read expressions. These skills will be developed). The focus is recognizing algebraic expressions versus numerical expressions.

Basic:

5x

3t-9

6 + m/n

m * 5n 4ab / 3g

Complex:

x(90 + 30)

2(81x + 9x -10)

2(3x - 4) -3(x - 5)

2(3(2x-4)-5) + Â½ (x+7) ñ 3/2

Ask students to provide more examples.

Any time two or more numbers, letters, or combination of letters and numbers are juxtaposed, it indicates multiplication. The number is called the coefficient of the variable. In algebraic expressions, a raised dot or parentheses are often used to indicate multiplication. Using "x" to indicate multiplication is avoided because it can be easily mistaken for the letter "x" (that is used as a variable). The following notations will be discussed in detail.

xy x ?y x(y) (x)y (x)(y)

In each expression, the quantities being multiplied are called factors, and the result is called the product. The above examples of algebraic expressions will now be verbalized.

Construct Algebraic Expression/Equations

It is important in the lesson to formulate the questions clearly and completely, and especially, to define the variables carefully and completely. The teacher should do this at the start, and gradually transfer the responsibility to the class, and then to each student. Therefore, although not shown in the unit due to space limitations, for each word problem, we will translate verbal descriptions into a verbal model and translate verbal model into a mathematical model or algebraic equation.

Verbal Description —> Verbal Model —> Algebraic Expression/Equation

Example 1:

Verbal Description: The sale price of a basketball is \$18. If the sale price is \$7 less than the original price, what is the original price?

Verbal Model: Sale Price = Original Price - Discount

Algebraic Equation: \$18 = Original Price - \$7

Example 2:

Verbal Description: The total income that an employee received in 1992 was \$21,550. Of that \$750 represented a bonus given at the end of the year. How much was the employee paid each week? Assume that each weekly paycheck contained the same amount, and that the year consisted of 52 weeks.

Verbal Model: Income for year = 52 times weekly pay + Bonus

Labels: Income for year = \$21,550

Weekly pay = x (in dollars)

Bonus = \$750

Algebraic Equation: 21,550 = 52x + 750

Problems

1) You are paid \$6 an hour. How much will you earn for working a certain number of hours? Let x represents hours. (6x)

2) A person is paid 3 cents for each aluminum soda can, and 2 cents for each steel soda can collected. What is the total amount the person will earn for the cans collected? (.03x + .02y)

3) A person is paid 3 cents for aluminum soda can, and 2 cents for each steel soda can collected, and \$45 a week for collecting other kinds of trash in the city park. (.03x + .02y + 45)

Write a verbal description for each of the following

1) 7x - 12 (Twelve less than the product of seven and a number)

2) 7(x-12) (Twelve is subtracted from a number and result is multiplied by seven).

It would be a good idea to have the students compare these two expressions. The second is equal to 72 les than the first.

3) (5+x)/2 (The sum of five and a number, all divided by two)

One of the primary goals of this unit is learning to read word problems and figure out what expressions and relations are needed to solve them. We need to emphasize to students that most word problems do not contain verbal expressions that clearly identify the arithmetic operations involved and that we need to sometimes rely on common sense and our experiences. This is an important skill, both for mathematics, and for making sense of many non-mathematical situations.

Example:

1) A cash register contains x quarters. Write an expression for this amount of money in dollars. (.25x)

2) A cash register contains n nickels and d dimes. Write an expression for this amount of money in cents. (5n + 10d)

# Lesson Two - Simplifying Expressions

The focus of this lesson will involve simplifying expressions. This will include using the distributive property including combining like terms within or without grouping symbols eliminating parentheses as much as possible. The order of operations will be reviewed.

Problem 1

Robert has 6 bags of red beads and 9 bags of green beads. How many beads are there in all the bags?

Here we assume that all the bags contain the same number of beads, which we will indicate by the unknown x. Then we may say that

Number of red beads = 6x

Number of green beads = 9x

Total number of beads = 6x + 9x = (6+9)x

= 15x

6x + 9x = x+x+x+x+x+x + x+x+x+x+x+x+x+x+x =15x

The terms 6x and 9x are called like terms because their variable portions are the same.

How many more green beads than red beads are there? 9x - 6x = (9 - 6)x = 3x

What factors do the addends have in common? Rewrite each sum.

2x + 7x = (2 + 7)x

= (9)x

= 9x

16x + 5x = (16 +5)x

= (21)x

=21x

12x - 30x = (12 -30)x

= (-18)x

=-18x

Note that these examples of "combining like terms" are all applications of the distributive property.

# Lesson Three - Translations of Verbal Descriptions (addition/subtraction)

The focus of this lesson is to translate verbal descriptions into numerical and algebraic expressions, then solve the expressions. We will participate in a whole class discussion introducing the topic and, if necessary, reviewing the common language list. This approach will provide opportunities for students to engage in active thinking processes, mathematics discourse, and problem-solving strategies.

In problem 1, the set of problems is progressing to the x + b = c form. Problem 1c requires students to add the two addends, where 1d requires students to find an addend and solve for n. Problems e and f, students must interpret expression verbally as well as written. It is imperative that the teacher take the time to define the variable carefully, and explain the reasoning involved in writing the relevant expressions or equations.

Problem 1

A number plus six (n + 6)

Problem 1b

A number plus six equals 32. (n + 6 = 32)

Problem 1c

Sara had \$26 in the bank. She made \$12 last week and put half of it the bank. What is the total amount of money Sara has in the bank. (26 + 1/2 (12) = 32)

Problem 1d

She made \$13 last week and put it the bank. She now has \$48 saved. How much money she Sara have originally? (Let n stands for the original amount) n + 13 = 48 (35)

Problem 1e - interpret and creates a word problem for the numerical equation

19 + 15 = 34

Problem 1f - interprets and creates a word problem for the numerical/symbolic equation

x + 27 = 40

Extension: The problems should become more difficulty as students learn how to develop this kind of reasoning.

Philadelphia has won 10 more games than it has lost. If they have lost 14 games, what is the total number of games played? (14 + (14 +10) = 14 + 24 = 38).

(If they have lost 14 games, and have won 10 games than that, hen they have won 14 + 10 = 24 games. The total number of games Philadelphia has played is the sum of the number they have won and the number they have lost. This is 14 + 24 = 38)

Problem 2 - Subtraction

In problem 2, the set of problems is progressing to the x - b = c form. Problem 2c requires students to add the two addends, where 2d requires students to find an addend and solve for n. Although these problems are possible and discounts are usually computed in percentages, I think doing these problems as indicated support and enhance the goals of the unit. In problem 2d, teacher should discuss the option of solving this problem such as using the distributive property. We want this skill to become routine to the students.

Problem 2a

A number is discounted from \$29.99. (29.99 - n)

Problem 2b

Amphone wants to buy a pair of jeans, which were originally \$29.99. The discount is \$4.75. How much do the jeans cost? (29.99 - 4.75 = 25.24)

Problem 2c

Marsha notices the original price of a calculator is \$13.99. The discount is 5.22. How much does she pay? (13.99 - 5.22 = 8.77)

Problem 2d

The difference between 8y and 2y is 42. What is y? (8y - 2y = (8 - 2)y = 6y = 42 (7))

# Lesson Four - Translations of Verbal Descriptions (multiplication/division)

In this lesson, a table will be used to emphasize writing algebraic expressions that uses multiplication and division. Students will then develop the skills to work with these problems abstractly.

Problem 2 (multiplication) - Concrete/Pictorial

There are 5 apples in a circle. How many apples are there total? (This will be a great time to assess/review students' understanding of expression notations.)

Number of Circles | Total Number of Apples

1 | 5*1 = 5

2 | 5*2 = 10

3 | 5*3 = 15

4 | 5*4 = 20

N | 5n, 5(n), 5*n

Abstraction: If n=8, how many apples are there altogether? If n=11, how many apples are there altogether. If n = 10, 12, 54, how many apples are there altogether?

Students will have verbalized and developed a concrete understanding that each time one circle is added, the total number of apples increases by five (repeated addition). They can now move to practicing the abstraction method. Try this!!!

More Abstraction with Multiplication!!

There are nine boxes of drumsticks. Each box contains p sticks.

Write the expression, in three ways, for the total number of sticks in terms of p.

Are there any differences with the three expressions? Explain.

Total number of sticks = 9p, 9-p, 9(p), (9)(p), 9*p

If each box contains 13 sticks, how many sticks are there altogether?

Problem 3

In problem 3, the set of problems are in the form of ax. Fractions are also used. It is still imperative that to formulate the questions clearly and completely, and especially, to define the variables carefully and completely. Students should be more comfortable in using in labeling variables. Problem 3b, the variable factor is the number of hours worked. Teachers may want to change the hours to different values and have the students calculate the earning. Problem 3d, students must find the missing factors and discussing and interpreting the answer as 6 Â½ cents per ounce will be beneficial. Also, in 3e, note the use of "altogether", in a situation where it does not mean either to add or to multiply.

Problem 3a

The product of a number and 10.50 ((10.50)n)

Problem 3b

You work at Burger King and recently received a raise. Your new pay rate is \$10.50. How much will you earn if you work 20 hours. (10.50(20) = 210)

Problem 3c

A plumber earns \$62 for each hour that she works. How much will she make for 30 hours? (62(30) = 1860)

Problem 3d

A marathon runner averages 10 miles per hour. How many miles did he run in 3 hours? 10(3) = 30

Problem 3d

A container of food weighing 22 ounces is purchased for \$1.43. Find the cost per ounce. (22x=1.43, .065)

Problem 3e

Two-thirds of Mrs. Minsinski's class are football fans. How many students are in the class if there are 16 football fans altogether? (2/3 * x = 16 (24))

Problem 3f

A box contains eight calculators. It is one-sixth full. If the box were full, how many calculators would be in the box? (1/6 * x = 8 (48))

Problem 3e - interpret the following equation.

6b = 54

Problem 3f - interpret the following equation

5/7m = 5

Problem 4 (Division)

Tyler has 4 boxes. He puts an equal number of CDs in each box. If there are 24 CDs, find the number of CDs in each box. If there are x CDs, find the number of CDs in each box in terms of x. (A discussion on "equal number" will be discussed; some examples will be given when the quantities are not divided equally).

1) 24/4 = 6

2) Number of CDs in each box = x/4

Is there a shorter way that the number of CDs in each box can be represented? Let c stand for the number of CDs in one box, let b stand for the number of boxes, and let t stand for the total number of CDs. Then,

3) cb = t or c = t/b

The following problems will serve as a review and reinforcement in writing and evaluating expressions.

Problem 4 - This set of problems investigates expressions of the form ax +b.

Problem 4a

Write an expression that means 14 times some number. (14x)

Problem 4b

Write an equation that says that 14 times some number plus 5 equals 33. (14x + 5 = 33)

Problem 4c

Suppose you made a 12-minute call from Denver to Atlanta. The call cost \$2.05 for the first three minutes and .34 for each additional minute.

(How much did the call cost? (.34(9) + 2.05= 5.11) Note: this is a good opportunity to talk about the fact that the total call was 12 minutes, and the base charge was for the first three minutes, so that the "additional minutes" were 12-3=9). The expression for an m minute call would be .34(x - 3) + 2.05.

Problem 4d

A sporting goods store uses a markup rate of 55% on all items. The cost to the store of a golf bag is \$35. What is the selling price of the bag? (35 + .55*35 = 54.25)

Problem 4e

Katie is going bowling. The bowling alley charges \$3 to rent shoes and \$1.50 for every game played. Katie needs to rent shoes. How many games can she play for \$25? 3 + 1.50x = 25 (â‰ˆ14). (In this problem, we will need to interpret a solution in the light of real world conditions that are not included in the equation.)

Problem 4f - -interpret the expression

.72 + .64x (A telephone call costs .72 cents for the first minute plus .64 cents for each additional minute)

# Lesson Five - Distributive Property

Sale Prices

Used Games \$9.95

Bargain Games \$14.95

Regular Games \$24.95

New Releases \$34.95

Instant Replay Video Games sells new and used games. During a Saturday morning sale, the first "x" customers each bought a bargain game and a new release. Write an expression to calculate the total sales for these customers.

14.95x + 34.95x or (14.95 + 34.95)x = 49.90x

Instant Replay Video Games sells new and used games. During a Saturday morning sale, the first 8 customers each bought a bargain game and a new release. What will be the total sales to these customers? (\$399.20)

Problem 1

The Morris family owns two cars. They drove the first car 18,000 miles and the second car 16,000 miles. Write an expression (using the distributive property) to calculate the cost to operate their cars in any year. Let x stand for the cost per mile to operate a car. Suppose that it is the same for the two cars.

18,000x + 16,000x = (18,000 + 16,000)x = 34,000x

Use the table to answer: The Morris family owns two cars. In 1998, they drove the first car 18,000 miles and the second car 16,000 miles. What will it cost the Morris family to operate their cars.

Car costs race ahead Prices (USA Today)

1985 .23

1990 .33

1995 .41

1998 .46

.46(18,000 + 16,000) = 15,640

Write an algebraic expression for each verbal expression. Then simplify, indicating the properties used.

1) twice the sum of s and t decreased by s.

2(s + t) -s

=2s + 2t - s (distributive rule)

=2s -s +2t (commutative rule)

= (2 - 1) s + 2t (distributive rule/combine like terms)

=s + 2t

2) five times the product of two and x increased by 3x

5(2x) + 3x (associative rule for multiplication)

=10x + 3x (combine like terms)

=13x (simplify)

# Lesson 6 - Combination - These problems combines all the components of the unit for reinforcement.

Problem 1

Roger has twice as much money as Elaine. Together they have \$84. How much does each have? (2x + x = 84 (Elaine: \$28, Roger: \$56))

Problem 1b

There are 850 students in Franklin High School. There are 30 more girls than boys. How many girls are there? ( x + (x+30) = 850 (440))

Expression Review - Write an expression for each verbal

1. Tyrone has some marbles. He puts x marbles in a bag. In all, he fills five bags, and then he has 3 marbles left over. How many marbles does Tyrone have?

Total number of marbles = 5x + 3

2. Jeff had \$50. He gave \$y to his son. The remainder was then shared equally between his two daughters. Express each daughter's share in terms of y.

Amount of money shared by the daughters = (50-y)/2

If y=12, how much money did each daughter receive? (19)

3. The admission fee to a bird park is \$y. The admission fee to an amusement park is \$1 more. Express the admission fee to the amusement park in terms of y. If the admission fee to the bird park is \$8, find the admission fee to the amusement park.

Admission Fee for Amusement Park = y + 1 (9)

# Lesson Seven - Application and Reinforcement

In lesson seven, students will continue to formulate notations and expressions as well as evaluate expressions. Students will work within small groups to expand and extend their knowledge of writing algebraic expressions to more complicated problems. These problems will require students to perform several embedded steps to demonstrate their proficiency. These problems should have elaborate discussions the first time students see it. Students will begin to solve more complex word problems such as:

Gabriele has been offered two summer jobs. The first job is a lifeguard position that will pay \$11 per hour but requires hat she take a \$200 certification class. The second job is a waitress position at a local restaurant. This job pays \$9 per hour, but she will have to buy a uniform for \$50. Gabrielle decides to make a few calculations concerning the money that can be made before deciding which job to take. Let x present the number of hours work. After how many hours will Gabrielle earn the same amount of money from both jobs? 11x - 200 = 9x - 50 (75)

A sum of \$10,000 is invested in two different type of accounts. Part of the money is invested in an account paying 9.5% simple interest, and the remainder is invested in an account paying 11% simple interest. At the end of the year, the two accounts pay a total interest of \$1,038.50. How much was invested in each account? (I = rP, where I is the interest, r is the annual interest rate, and p is the principal).

Both accounts: interest \$1,038.50 principal = 10,000

First account: interest rate = .095 principal = x

Second account: interest rate = .11. principal = 10,000 - x

Equation: .095x + .11(10,000 - x) = 1,038.50 (\$4100 at 9.5% & \$5,900 at 11%)

Assessment

The unit will include informal and formal assessments. Observations will be done during classroom discussions: whole and small groups. Unit assignments, quizzes, and test will be conducted and written and verbal feedback will be provided. These assessments will direct the pace as well as the lessons.

# Appendix 1 - ALGEBRAIC OPERATIONS

"Common Language"

Augment

Combine

Enlarge

Exceeds

Gain

Greater than

Grow

In all

Increase by

Larger than

More than

Plus

Rise

Sum

Total

Subtraction

decreased

deduct

depreciate

difference

diminish

drop

fewer

left

less than

lose

loss

lower

minus

remain

remove

smaller than

Subtract

Take away

Multiplication

cubed

double

factor

multiple of

multiply

squared

times

triple

twice

Division

average

(a) fourth

divided by

equally

half

per

quotient

ratio

shared

split

(a) third

Adopted from The Math Teacher's Book of List

Properties of Operations with Real Numbers:

Commutative Property of Addition a + b = b + a

Commutative Property of Multiplication ab = ba

Associative Property of Addition (a + b ) + c = a + (b + c)

Associative Property of Multiplication (ab)c = a(bc)

Additive Identity Property a + 0 = 0 + a

Multiplicative Identify Property a* 1 = 1 * a = a

Additive Inverse Property a + (-a) = 0

Multiplicative Inverse Property a * (1/a) = 1, a â‰  0

Distributive Property a(b + c) = ab + ac

# BIBLIOGRAPHY FOR TEACHERS

Aharoni, Ron. (2007). Arithmetic for Parents: A Book for Grownups About Children's

Mathematics. El Cerrito, CA: Sumizdat.

The book provided detailed discussions on basic elementary mathematics topics and shows how grownups can use this information to develop and/or improve their mathematical thinking.

Burns, Marilyn. (1998). Math: Facing an American Phobia. Sausalito, CA: Math

Solution Publications.

This book looks at why math has the dreadful reputation it does. It provides clear messages about what math can and should mean to us all and how we can keep our children from developing negative attitudes. Also, it is an excellent resource for parents.

Burns, Marilyn. (1984). The Math Solution: Teaching Mastery through Problem Solving.

Sausalito, CA: Math Solution Publications.

This book provides strategies and examples of problem solving techniques.

Carnegie Learning. (2006). Algebra I Student Text. Pittsburgh, PA: Carnegie Learning,

Inc.

Writing and evaluating algebraic expressions are clearly explained in Chapters 1-3. Many practice exercises and illustrations are included.

Hogan, Bob & Forsten, Char. (2007). 8-Step Model Drawing. Peterborough, NW:

Crystal Springs Books.

This book introduces the model-drawing process adapted from the Singapore Math. Many of the word problems for the units can from this book - excellent book for teachers who are interesting in using the Singaporemath curriculum.

Larson, Roland E. & Hostetler, Robert P. (1992). Elementary Algebra. Lexington, MA:

D.C. Heath.

Many of the word problems used in the unit came from this textbook.

Ma, Liping. (1999). Knowing and Teaching Elementary Mathematics. Lawrence

Earlbaum Associates, Inc. New Jersey.

This text was used a reference and the ideas was discussed with the seminar.

Marzano, Robert J. (2007). The Art and Science of Teaching. Alexandria, VA: Association of Supervision and Curriculum Development.

This text presented a model for ensuring quality teaching that balances the necessity of research-based instructional strategies and design.

Algebra I Student Textbook. (2007). Glencoe Mathematics. McGraw Hill, Chicago, IL.

A textbook designed for use in a year long algebra course in linear functions. Writing and evaluating algebraic expressions are clearly explained in Chapters 1-3. Many practice exercises and illustrations are included. Many word problems in this unit came from this text.

Marshall Cavendish Education, Primary Mathematics Workbook 5A. (1984).

SingaporeMath. Oregon City, OR.

This text was used as a reference. Many word problems in this unit came from this text.

Marshall Cavendish Education, Primary Mathematics Workbook 5B. (1984).

SingaporeMath. Oregon City, OR.

This text was used as a reference. Many word problems in this unit came from this text.

Marshall Cavendish Education, Primary Mathematics Workbook 6A. (1984).

SingaporeMath. Oregon City, OR.

This text was used as a reference. Many word problems in this unit came from this text.

www.cps.k12.il.us. News clipping. Retrieve July 4, 2007

This website provided information regarding the school district and the school.

# STUDENTS RESOUCRES

Each of these books was reviewed and discussed in the seminar. It was agreed that they will be excellent resources for teachers as well as students. They support literacy while applying mathematical skills.

Anno, Mitsumasa. (1995). Anno's Magic Seeds. New York: Philomel Books.

A wizard gives Jack two golden seeds and directs him to eat one and bury the other. He promises it will grow and give 2 more magic seeds in the fall. Jack does as he is told, and the cycle repeats for a number of years, until Jack decides to bury both seeds. The tale of exponential growth is discovered as Jack buries more and more seeds. The math tale becomes even more rigorous as Jack marries, has a child, begins to store some seeds and sell others... until a hurricane wipes out the crops and Jack must begin all over again.

Axelrod. Amy. (1994). Pigs will be Pigs: Fun with Math and Money. New York: Simon

and Schuster Books for Young Readers.

After gobbling up all the groceries, Mr. Pig, Mrs. Pig and their two piglets are hungry again, but the Piggy bank is empty. The Pigs turn their house upside down looking for spare change so that they can go out to dinner. Readers are meant to keep a tally of the dimes and nickels the Pigs locate. Finally, after finding a grand total of \$34.67, the Pigs spend almost all of it at a Mexican restaurant and readers can calculate the tab by reading a menu.

Burns, Marilyn. (1997). Spaghetti and Meatballs for All: A Mathematical Story. New

York: Scholastic.

Mrs. and Mr. Comfort are throwing a dinner party for their family. They find out there will be 32 people. They arrange 8 tables with 4 chairs each. The problem arises when the families begin to arrive and want to sit together. They begin pushing the tables together to make one big table. As more and more guests arrive, the families continue to rearrange the tables so everyone can sit together. Of course throughout this time Mrs. Comfort is getting very distraught because she knows that if the tables are pushed together there will not be a seat for everyone. The story continues with more arrangements of the tables until eventually the tables start being pulled apart as the rest of the family arrives. In the end the tables again end up as 8 tables with 4 chairs each, just as Mrs. Comfort had originally set.

Demi. (1997). One Grain of Rice: A Mathematical Folktale. New York: Scholastic.

It's the story of Rani, a clever girl who outsmarts a very selfish raja and saves her village. When offered a reward for a good deed, she asks only for one grain of rice, doubled each day for 30 days. That's lots of rice: enough to feed a village for a good long time—and to teach a greedy raja a lesson.

Lewis, J. Patrick. (2002). Arithmetickle: An Even Number of Odd Riddle-Rhymes. San

Diego: Harcourt.

This book offers a variety of clever math riddles with titles like "Finger Play" (which teaches a nifty trick for multiplying by nine) and "Your Average Cow," which asks kids to compare bovine and human life expectancies. Answers appear (upside-down) below each entry.

# Classroom Material

Graphing Calculators

Transparencies

Easel Paper

Colored Pencils

Markers

# Implementing District Standards

Standard 8A - Students who meet the standard can describe numerical relationships using

Variables and patterns.

Students will evaluation algebraic expressions for given values and simplify algebraic expressions involving like terms. Students experience with numbers and patterns will form a foundation for symbols, properties, and algebraic expressions.

Standard 8B - Students who meet the standard can interpret and describe numerical

relationships using tables, graphs, and symbols.

Students must be able to use algebraic methods to construction expressions and equations. Students must be able to interpret the relationships expressed by expressions and equations.

Standard 8C - Students who meet the standard can solve problems using systems of

numbers and their properties.

Students must be able to find solutions to everyday problems using the tools of algebra and logic.